Graphs with tiny vector chromatic numbers and huge chromatic numbers

Karger Motwani and Sudan (1998) introduced the notion of a vector coloring of a graph. In particular they show that every k-colorable graph is also vector k-colorable, and that for constant k, graphs that are vector k-colorable can be colored by roughly Δ^1-2k/ colors. Here Δ is the maximum degree in the graph. Their results play a major role in the best approximation algorithms for coloring and for maximal independent set. We show that for every positive integer k there are graphs that are vector k-colorable but do not have independent sets significantly larger than n/Δ^1-2k/ (and hence cannot be colored with significantly less that Δ^1-2k/ colors). For k = O(log n/log log n) we show vector k-colorable graphs that do not have independent sets of size (log n)^c, for some constant c. This shows that the vector chromatic number does not approximate the chromatic number within factors better than n/polylogn. As part of our proof, we analyze "property testing" algorithms that distinguish between graphs that have an independent set of size n/k, and graphs that are "far" from having such an independent set. Our bounds on the sample size improve previous bounds of Goldreich, Goldwasser and Ron (1998) for this problem.